Owain Hughes

As a Welsh person I didn't get this either but have been told today what it means and it is nothing to do with Owing anyone money.
Apparently the verb "to hew" means to cut something down with an axe. So Owain Hughes is Owain who cuts down things with an axe. That why he said himself "before you ask no I don't" and why Uncle Bryn said "and do you" when he was introduced to Owain.

Isn't funny when it's explained in detail and I think it wasn't meant to be explained, thats what made it funny.